Sutured Manifolds Research Articles

Morse–Novikov theory, Heegaard splittings, and closed orbits of gradient flows

The works of Donaldson (2) and Mark (14) make the structure of the Seiberg-Witten invariant of 3-manifolds clear. It corresponds to certain torsion type invariants counting flow lines and closed orbits of a gradient flow of a circle- valued Morse map on a 3-manifold. We study these invariants using the Morse- Novikov theory and Heegaard splitting for sutured manifolds, and make detailed computations for knot complements.

Seifert surfaces distinguished by sutured Floer homology but not its Euler characteristic

In this paper we find a family of knots with trivial Alexander polynomial, and construct two non-isotopic Seifert surfaces for each member in our family. In order to distinguish the surfaces we study the sutured Floer homology invariants of the sutured manifolds obtained by cutting the knot complements along the Seifert surfaces. Our examples provide the first use of sutured Floer homology, and not merely its Euler characteristic (a classical torsion), to distinguish Seifert surfaces. Our technique uses a version of Floer homology, called “longitude Floer homology” in a way that enables us to bypass the computations related to the SFH of the complement of a Seifert surface.

The virtual fibering theorem for 3-manifolds

In 2007 Agol showed that if N is an aspherical compact 3-manifold with empty or toroidal boundary such that \pi_1(N) is virtually RFRS, then N is virtually fibered. We give a largely self-contained proof of Agol’s theorem using complexities of sutured manifolds.

The sutured Floer polytope and taut depth-one foliations

For closed 3-manifolds, Heegaard Floer homology is related to the Thurston norm through results due to Ozsv\'ath and Szab\'o, Ni, and Hedden. For example, given a closed 3-manifold Y, there is a bijection between vertices of the HF^+(Y) polytope carrying the group Z and the faces of the Thurston norm unit ball that correspond to fibrations of Y over the unit circle. Moreover, the Thurston norm unit ball of Y is dual to the polytope of \underline{\hfhat}(Y). We prove a similar bijection and duality result for a class of 3-manifolds with boundary called sutured manifolds. A sutured manifold is essentially a cobordism between two surfaces R_+ and R_- that have nonempty boundary. We show that there is a bijection between vertices of the sutured Floer polytope carrying the group Z and equivalence classes of taut depth one foliations that form the foliation cones of Cantwell and Conlon. Moreover, we show that a function defined by Juh\'asz, which we call the geometric sutured function,is analogous to the Thurston norm in this context. In some cases, this function is an asymmetric norm and our duality result is that appropriate faces of this norm's unit ball subtend the foliation cones. An important step in our work is the following fact: a sutured manifold admits a fibration or a taut depth one foliation whose sole compact leaves are exactly the connected components of R_+ and R_-, if and only if, there is a surface decomposition of the sutured manifold resulting in a connected product manifold.

Exceptional surgeries on knots with exceptional classes

We survey the aspects of classical combinatorial sutured manifold theory and show how they can be adapted to study exceptional Dehn fillings and 2-handle additions. As a consequence, we show that if a hyperbolic knot \(\beta \) in a compact, orientable, hyperbolic 3-manifold, \(M\) has the property that winding number and wrapping number are not equal with respect to a non-trivial class in \(H_2(M,\partial M)\), and then, with only a few possible exceptions, every 3-manifold \(M'\) obtained by Dehn surgery on \(\beta \) with surgery distance \(\Delta \ge 2\) will be hyperbolic.

Comparing 2-handle additions to a genus 2 boundary component

We prove that knots obtained by attaching a band to a split link satisfy the cabling conjecture. We also give new proofs that unknotting number one knots are prime and that genus is superadditive under band sum. Additionally, we prove a collection of results comparing two 2-handle additions to a genus two boundary component of a compact, orientable 3-manifold. These results give a near complete solution to a conjecture of Scharlemann and provide evidence for a conjecture of Scharlemann and Wu. The proofs make use of a new theorem concerning the effects of attaching a 2-handle to a suture in the boundary of a sutured manifold.

Band-taut sutured manifolds

Attaching a 2-handle to a genus two or greater boundary component of a 3-manifold is a natural generalization of Dehn filling a torus boundary component. We prove that there is an interesting relationship between an essential surface in a sutured 3-manifold, the number of intersections between the boundary of the surface and one of the sutures, and the cocore of the 2-handle in the manifold after attaching a 2-handle along the suture. We use this result to show that tunnels for tunnel number one knots or links in any 3-manifold can be isotoped to lie on a branched surface corresponding to a certain taut sutured manifold hierarchy of the knot or link exterior. In a subsequent paper, we use the theorem to prove that band sums satisfy the cabling conjecture, and to give new proofs that unknotting number one knots are prime and that genus is superadditive under band sum. To prove the theorem, we introduce band taut sutured manifolds and prove the existence of band taut sutured manifold hierarchies.

Some applications of Gabaiʼs internal hierarchy

Any Haken 3-manifold (possibly with boundary consisting of tori) can be transformed into a surface×S1 by a series of splitting and regluing along incompressible surfaces. This fact was proved by Gabai as an application of his sutured manifold theory. The first half of this paper provides a few technical details in the proof. In the second half of this paper, some applications of Gabaiʼs theorem to Heegaard Floer homology are given. We refine the known results about the Thurson norm and fibrations. We also give some classification results for Floer simple knots in manifolds with positive b1.

Homology Cylinders and Sutured Manifolds for Homologically Fibered Knots

Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by focusing on sutured manifolds associated with a special class of knots which we call {\it homologically fibered knots}. Then we use invariants of homology cylinders to give applications to knot theory such as fibering obstructions, Reidemeister torsions and handle numbers of homologically fibered knots.

SUTURED TQFT, TORSION AND TORI

We use the theory of sutured TQFT to classify contact elements in the sutured Floer homology, with ℤ coefficients, of certain sutured manifolds of the form (Σ × S1, F × S1) where Σ is an annulus or punctured torus. Using this classification, we give a new proof that the contact invariant in sutured Floer homology with ℤ coefficients of a contact structure with Giroux torsion vanishes. We also give a new proof of Massot's theorem that the contact invariant vanishes for a contact structure on (Σ × S1, F × S1) described by an isolating dividing set.

On sutured Floer homology and the equivalence of Seifert surfaces

We study the sutured Floer homology invariants of the sutured manifold obtained by cutting a knot complement along a Seifert surface, R. We show that these invariants are finer than the "top term" of the knot Floer homology, which they contain. In particular, we use sutured Floer homology to distinguish two non-isotopic minimal genus Seifert surfaces for the knot 8_3. A key ingredient for this technique is finding appropriate Heegaard diagrams for the sutured manifold associated to the complement of a Seifert surface.

Taut sutured manifolds and twisted homology

We give a necessary and sufficient criterion for a sutured manifold (M,γ) to be taut in terms of the twisted homology of the pair (M,R−).

Monopole Floer homology and Legendrian knots

We use monopole Floer homology for sutured manifolds to construct invariants of Legendrian knots in a contact 3-manifold. These invariants assign to a knot K in Y elements of the monopole knot homology KHM(-Y,K), and they strongly resemble the knot Floer homology invariants of Lisca, Ozsv\'ath, Stipsicz, and Szab\'o. We prove several vanishing results, investigate their behavior under contact surgeries, and use this to construct many examples of non-loose knots in overtwisted 3-manifolds. We also show that these invariants are functorial with respect to Lagrangian concordance.

Sutured Floer homology and hypergraphs

By applying Seifert's algorithm to a special alternating diagram of a link L, one obtains a Seifert surface F of L. We show that the support of the sutured Floer homology of the sutured manifold complementary to F is affine isomorphic to the set of lattice points given as hypertrees in a certain hypergraph that is naturally associated to the diagram. This implies that the Floer groups in question are supported in a set of Spin^c structures that are the integer lattice points of a convex polytope. This property has an immediate extension to Seifert surfaces arising from homogeneous link diagrams (including all alternating and positive diagrams). In another direction, together with work in progress of the second author and others, our correspondence suggests a method for computing the "top" coefficients of the HOMFLY polynomial of a special alternating link from the sutured Floer homology of a Seifert surface complement for a certain dual link.

Sutured Floer homology, sutured TQFT and noncommutative QFT

We define a "sutured topological quantum field theory", motivated by the study of sutured Floer homology of product 3-manifolds, and contact elements. We study a rich algebraic structure of suture elements in sutured TQFT, showing that it corresponds to contact elements in sutured Floer homology. We use this approach to make computations of contact elements in sutured Floer homology over $\Z$ of sutured manifolds $(D^2 \times S^1, F \times S^1)$ where $F$ is finite. This generalises previous results of the author over $\Z_2$ coefficients. Our approach elaborates upon the quantum field theoretic aspects of sutured Floer homology, building a non-commutative Fock space, together with a bilinear form deriving from a certain combinatorial partial order; we show that the sutured TQFT of discs is isomorphic to this Fock space.

Unknotting genus one knots

For any knot with genus one and unknotting number one, other than the figure-eight knot, we prove that there is exactly one way to unknot it by means of a crossing change. In the case of the figure-eight knot, we prove that there are precisely two unknotting crossing changes. The proof uses sutured manifold theory and an analysis of the arc complex of the once-punctured torus.

The decategorification of sutured Floer homology

We define a torsion invariant T for every balanced sutured manifold (M,g), and show that it agrees with the Euler characteristic of sutured Floer homology SFH. The invariant T is easily computed using Fox calculus. With the help of T, we prove that if (M,g) is complementary to a Seifert surface of an alternating knot, then SFH(M,g) is either 0 or Z in every spin^c structure. T can also be used to show that a sutured manifold is not disk decomposable, and to distinguish between Seifert surfaces. The support of SFH gives rise to a norm z on H_2(M, \partial M; R). Then T gives a lower bound on the norm z, which in turn is at most the sutured Thurston norm x^s. For closed three-manifolds, it is well known that Floer homology determines the Thurston norm, but we show that z < x^s can happen in general. Finally, we compute T for several wide classes of sutured manifolds.

The sutured Floer homology polytope

In this paper, we extend the theory of sutured Floer homology developed by the author. We first prove an adjunction inequality, and then define a polytope P(M,g) in H^2(M,\partial M; R) that is spanned by the Spin^c-structures which support non-zero Floer homology groups. If (M,g) --> (M',g') is a taut surface decomposition, then a natural map projects P(M',g') onto a face of P(M,g); moreover, if H_2(M) = 0, then every face of P(M,g) can be obtained in this way for some surface decomposition. We show that if (M,g) is reduced, horizontally prime, and H_2(M) = 0, then P(M,g) is maximal dimensional in H^2(M,\partial M; R). This implies that if rk(SFH(M,g)) < 2^{k+1} then (M,g) has depth at most 2k. Moreover, SFH acts as a complexity for balanced sutured manifolds. In particular, the rank of the top term of knot Floer homology bounds the topological complexity of the knot complement, in addition to simply detecting fibred knots.

On the Naturality of the Spectral Sequence from Khovanov Homology to Heegaard Floer Homology

In [18], Ozsvath-Szabo established an algebraic relationship, in the form of a spectral sequence, between the reduced Khovanov homology of (the mirror of) a link and the Heegaard Floer homology of its double-branched cover. This relationship, extended in [19] and [4], was recast, in [5], as a specific instance of a broader connection between Khovanov- and Heegaard Floer-type homology theories, using a version of Heegaard Floer homology for sutured manifolds developed by Juhasz in [7]. In the present work, we prove the naturality of the spectral sequence under certain elementary operations, using a generalization of Juhasz’s surface decomposition theorem valid for decomposing surfaces geometrically disjoint from an imbedded framed link.

Knots, sutures, and excision

We develop monopole and instanton Floer homology groups for balanced sutured manifolds, in the spirit of A. Juhasz. Holomorphic discs and sutured manifolds. Algebr. Geom. Topol., 6:1429–1457 (electronic), 2006. Applications include a new proof of Property P for knots.